E ENGINEERING
WWII: Liberty ships Reprinted w/ permission from R.W. Hertzberg, "Deformation and Fracture Mechanics of Engineering Materials", (4th ed.) Fig. 7.1(b), p. 6, John Wiley and Sons, Inc., 1996. (Orig. source: Earl R. Parker, "Behavior of Engineering Structures", Nat. Acad. Sci., Nat. Res. Council, John Wiley and Sons, Inc., NY, 1957.)
Course Content: A - INTRODUCTION Mechanical failure modes; Review of load and stress analysis equilibrium equations, complex stresses, stress transformation, Mohr s circle, stress-strain relations, stress concentration; Fatigue design methods; Design strategies; Design criteria. B MATERIALS ASPECTS OF FATIGUE AND FRACTURE Static fracture process; Fatigue fracture surfaces; Macroscopic features; Fracture mechanisms; Microscopic features. C FATIGUE: STRESS-LIFE APPROACH Fatigue loading; Fatigue testing; S-N curve; Fatigue limit; Mean stress effects; Factors affecting S-N behavior microstructure, size effect, surface finish, frequency. 3
D FATIGUE: STRAIN-LIFE APPROACH Stress-strain diagram; Strain-controlled test methods; Cyclic stress-strain behavior; Strain-based approach to life estimation; Strain-life fatigue properties; Mean stress effects; Effects of surface finish. E LINEAR ELASTIC Fundamentals of LEFM loading modes, stress intensity factor, K; Geometry correction factors; Superposition for Mode I; Crack-tip plasticity; Fracture toughness, K IC ; Plane stress versus plane strain fracture; Extension to elastic-plastic fracture.. F FATIGUE CRACK PROPAGATION Fatigue crack growth; Paris Law; da/dn- K; Crack growth test method; Threshold K th ; Mean stress effects; Crack growth life integration. 4
Mechanics of Materials A branch of mechanics that studies the relationships between external loads applied to a deformable body and the intensity of internal forces acting within the body. Fracture Mechanics The mechanics that describes the response of materials to loading in the presence of crack or crack-like defects.
A Short Course in Fracture Mechanics (Typical course content) Introduction Historical Review Fracture mechanics approaches Linear Elastic Fracture Mechanics Elastic stress field approach Crack tip plasticity Energy balance approach LEFM testing Elastic-Plastic Fracture Mechanics J-integral COD approach Fracture Mechanics Concept for Crack Growth Fatigue crack growth Dynamic crack growth and arrest Time-Dependent Fracture Fracture Mechanisms in Metals and Nonmetals
References: Anderson, T.L., Fracture Mechanics Fundamentals and Applications, 3 rd edition, CRC Press, FL, USA, 005. Broek, D., Elementary Engineering Fracture Mechanics, KluwerAcademic Publishers, 1991. Atkins, A.G. and Mai, Y.W., Elastic and Plastic Fracture Metals, Polymers, ceramics, Composites, Biological Materials, Ellis Horwood Ltd., UK, 1985. 7
Linear Elastic Fracture Mechanics (LEFM) Fracture mechanics within the confines of the theory of linear elasticity. Analytical procedure that relates the stress magnitude and distribution in the neighborhood of a crack to: the nominal applied stress crack geometry (size, shape) and orientation material properties An underlying principle is that unstable fracture occurs when the stress-intensity factor at the crack tip reaches a critical value.
Scope of fracture mechanics
Basic loading of cracked bodies
Stress field ahead of crack tip = 3 sin sin 1 cos θ θ θ σ σ r a x + = 3 sin sin 1 cos θ θ θ σ σ r a y r θ Westergaard solution = 3 cos sin cos θ θ θ σ τ r a xy K I is called stress intensity factor (SIF) ( ) ( ) terms order higher f r K ij I ij + = θ π σ
Stress intensity factor limσij = 0 r K I πr f ij ( ) θ σ σ yy τ xy The stress intensity factor, K I describes the crack tip stresses. Crack r θ σ xy K I = βσ a a β- dimensionless parameter K I has dimension of MPa m σ For infinite cracked plate K I = σ πa
Stress field at notch tip Compact tension C(T) specimen σ yy von Mises
Crack-tip stress r K r K I y I y π σ θ θ θ π σ 3 sin sin 1 cos = + =
Crack-tip plasticity σ = YS K I * πrp r K σ a * I p = = πσys σys
Shape of plastic zone ( ) + + = θ θ σ π θ sin 3 cos 1 4 1 YS I y K r 16 ( ) ( ) ( ) + + = θ θ ν σ π θ sin 3 cos 1 1 4 1 YS I y K r Plane strain
Finite width correction σ Meaningful parameters are σ and a K I = σ πa Infinite cracked plate K I = Yσ πa Y = πa sec w σ 17
Finite width correction for SIF For a<<w, K I = σ πa
SIF
SIF
Condition for fracture Fracture occurs when the applied stress intensity factor, K I reaches the value of the fracture toughness, K IC of the material KI K IC σ K I =1.1σ πa a At fracture: 1.1σ c πa = K IC 1.1σ πa c = K IC
Fracture Toughness 0.5) KIc(MPa m 0 100 70 60 50 40 30 0 10 7 6 5 4 3 Metals/ Alloys Steels Ti alloys Al alloys Mg alloys Graphite/ Ceramics/ Semicond Diamond Si carbide Al oxide Si nitride Polymers PET PP PVC PC Composites/ fibers C-C( fibers) 1 Al/Al oxide(sf) Y O 3 /ZrO (p) 4 C/C( fibers) 1 Al oxid/sic(w) 3 Si nitr/sic(w) 5 Al oxid/zro (p) 4 Glass/SiC(w) 6 Fracture toughness represents the resistance of materials to resist cracking. Fracture toughness values are determined from fracture toughness tests. 1 0.7 0.6 0.5 <100> Si crystal <111> Glass-soda Concrete PS Polyester Glass 6 Based on data in Table B5, Callister 6e.
Effect of thickness Ref. : T.L Anderson, CRC Press, 005 Plane stress Plane strain
Plane Stress versus Plane Strain
Plane stress versus plane strain fracture Ref. : Atkins and Mai, 1985 5
Stress triaxiality at crack-tip Ref. : T.L Anderson, CRC Press, 005 6
Residual strength
K-controlled fracture K I characterizes crack-tip condition even though the 1/ r singularity does not apply to the plastic zone. LEFM ceased to be valid when the plastic zone size becomes large relative to key dimensions 8
Design considerations Case I - Maximum flaw size dictates the design stress. σ design < K c Y πa max σ Case II -Design stress dictates the tolerable maximum flaw size. a max < 1 K c π Yσ design no fracture fracture amax
Energy release rate Change in energy, dudue to crack growth from a to a+dais represented by the shaded area. du = da 0 σ y u y dr 30
Energy release rate u r du = y da 0 σ σ = σ y u y a r dr ( ν ) a ( da r) = 1 E σ du σ πa = E ( 1 ν )da Energy release rate, G du da σ πa = E ( 1 ν ) 31
Energy release rate G = du da σ πa = E ( 1 ν ) Plane strain G σ πa E = Plane stress G = E K ( 1 ν ) Plane strain G = K E Plane stress 3
Fracture Toughness Test ASTM E399 Standard Test Method for Plane Strain Fracture Toughness of Metallic Materials Sample geometry 33
Fracture Toughness Test Direction of cut 34
Fracture Toughness Test Fatigue pre-cracking K = P B W f a W 35
Fracture Toughness Test Load-gage displacement K Q = P Q B W f a W Validity requirements for K IC 0.45 a W K B, a.5 σ P 1.10P max 0.55 YS Q Q 36
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